Sources page biographical material

INDEX

( 3, 13) Dudeney, Stong

( 3, 15) Mittenzwey, Hoffmann, Mr. X, Dudeney, Blyth,

( 3, 17) Fourrey,

( 3, 21) Blyth, Hummerston,

( 4, 15) Mittenzwey,

( 6, 30) Pacioli, Leske, Mittenzwey, Ducret,

( 6, 31) Baker,

( 6, 50) Ball-FitzPatrick,

( 6, 52) Rational Recreations

( 6, 57) Hummerston,

( 7, 40) Mittenzwey,

( 7, 41) Sprague,

( 7, 45) Mittenzwey,

( 7, 50) Decremps,

( 7, 60) Fourrey,

( 8, 100) Bachet, Carroll,

( 9, 100) Bachet, Ozanam, Alberti

(10, 100) Bachet, Henrion, Ozanam, Alberti, Les Amusemens, Hooper, Decremps,

Badcock, Jackson, Rational Recreations, Manuel des Sorciers,

Boy's Own Book, Nuts to Crack, Young Man's Book, Carroll,

Magician's Own Book, Book of 500 Puzzles, Secret Out,

Boy's Own Conjuring Book, Vinot, Riecke, Fourrey, Ducret, Devant,

(10, 120) Bachet,

(12, 134) Decremps,

General case: Bachet, Ozanam, Alberti, Decremps, Boy's Own Book, Young Man's Book, Vinot, Mittenzwey, (others ?? check)

Versions with limited numbers of each value or using a die -- see 4.A.1.a.

Version where an odd number in total has to be taken: Dudeney, Grossman & Kramer, Sprague.

Versions with last player losing: Mittenzwey,

Pacioli. De Viribus. c1500. Ff. 73v - 76v. XXXIIII effecto afinire qualunch' numero na'ze al compagno anon prendere piu de un termi(n)ato .n. (34th effect to finish whatever number is before the company, not taking more than a limiting number) = Peirani 109 112. Phrases it as an addition problem. Considers (6, 30) and the general problem.

David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games. Penguin, 1991, pp. 174-175. "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)."

Cardan. Practica Arithmetice. 1539. Chap. 61, section 18, ff. T.iiii.v - T.v.r (p. 113). "Ludi mentales". One has 1, 3, 6 and the other has 2, 4, 5; or one has 1, 3, 5, 8, 9 and the other has 2, 4, 6, 7, 10; one one wants to make 100. "Sunt magnæ inventionis, & ego inveni æquitando & sine aliquo auxilio cum socio potes ludere & memorium exercere ...."

Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670: pp. 353 354. ??NX. (6, 31).

Bachet. Problemes. 1612. Prob. XIX: 1612, 99-103. Prob. XXII, 1624: 170-173; 1884: 115 117. Phrases it as an addition problem. First considers (10, 100), then (10, 120), (8, 100), (9, 100), and the general case. Labosne omits the demonstration.

Dennis Henrion. Nottes to van Etten. 1630. Pp. 19-20. (10, 100) as an addition problem, citing Bachet.

Ozanam. 1694. Prob. 21, 1696: 71-72; 1708: 63 64. Prob. 25, 1725: 182 184. Prob. 14, 1778: 162-164; 1803: 163-164; 1814: 143-145. Prob. 13, 1840: 73-74. Phrases it as an addition problem. Considers (10, 100) and (9, 100) and remarks on the general case.

Alberti. 1747. Due persone essendo convenuto ..., pp. 105 108 (66 67). This is a slight recasting of Ozanam.

Les Amusemens. 1749. Prob. 10, p. 130: Le Piquet des Cavaliers. (10, 100) in additive form. "Deux amis voyagent à cheval, l'un propose à l'autre un cent de Piquet sans carte."

William Hooper. Rational Recreations, In which the Principles of Numbers and Natural Philosophy Are clearly and copiously elucidated, by a series of Easy, Entertaining, Interesting Experiments. Among which are All those commonly performed with the cards. [Taken from my 2nd ed.] 4 vols., L. Davis et al., London, 1774; 2nd ed., corrected, L. Davis et al., London, 1783-1782 (vol. 1 says 1783, the others say 1782; BMC gives 1783-82); 3rd ed., corrected, 1787; 4th ed., corrected, B. Law et al., London, 1794. [Hall, BCB 180-184 & Toole Stott 389-392. Hall says the first four eds. have identical pagination. I have not seen any difference in the first four editions, except as noted in Section 6.P.2. Hall, OCB, p. 155. Heyl 177 notes the different datings of the 2nd ed, Hall, BCB 184 and Toole Stott 393 is a 2 vol. 4th ed., corrected, London, 1802. Toole Stott 394 is a 2 vol. ed. from Perth, 1801. I have a note that there was an 1816 ed, but I have no details. Since all relevant material seems the same in all volumes, I will cite this as 1774.] Vol. 1, recreation VIII: The magical century. (10, 100) in additive form. Mentions other versions and the general rule.

I don't see any connection between this and Rational Recreations, 1824.

Henri Decremps. Codicile de Jérôme Sharp, Professeur de Physique amusante; Où l'on trouve parmi plusieurs Tours dont il n'est point parlé dans son Testament, diverses récréations relatives aux Sciences & Beaux-Arts; Pour servir de troisième suite À La Magie Blanche Dévoilée. Lesclapart, Paris, 1788. Chap. XXVII, pp. 177-184: Principes mathématiques sur le piquet à cheval, ou l'art de gagner son diner en se promenant. Does (10, 100) in additive form, then discusses the general method, illustrating with (7, 50) and (12, 134).

Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33-34, no. 48: A curious recreation with a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50 counters. Discusses general case, but doesn't notice that the limitation to 50 counters each considerably changes the game!

Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 47, pp. 11 & 64. Additive form of (10, 100).

Rational Recreations. 1824. Exercise 12(?), pp. 57-58. As in Badcock. Then says it can be generalised and gives (6, 52).

Manuel des Sorciers. 1825. Pp. 57-58, art. 30: Le piquet sans cartes. ??NX (10, 100) done subtractively.

The Boy's Own Book.

The certain game. 1828: 177; 1828-2: 236; 1829 (US): 104; 1855: 386 387; 1868: 427.

The magical century. 1828: 180; 1828-2: 236 237; 1829 (US): 104-105; 1855: 391 392.

Both are additive phrasings of (10, 100). The latter mentions using other numbers and how to win then.

Nuts to Crack V (1836), no. 70. An arithmetical problem. (10, 100).

Young Man's Book. 1839. Pp. 294-295. A curious Recreation with a Hundred Numbers, usually called the Magical Century. Almost identical to Boy's Own Book.

Lewis Carroll.

Diary entry for 5 Feb 1856. In Carroll-Gardner, pp. 42-43. (10, 100). Wakeling's note in the Diaries indicates he is not familiar with this game.

Diary entry for 24 Oct 1872. Says he has written out the rules for Arithmetical Croquet, a game he recently invented. Roger Lancelyn Green's abridged version of the Diaries, 1954, prints a MS version dated 22 Apr 1889. Carroll-Wakeling, prob. 38, pp. 52-53 and Carroll-Gardner, pp. 39 & 42 reprint this, but Gardner has a misprinted date of 1899. Basically (8, 100), but passing the values 10, 20, ..., requires special moves and one may have to go backward. Also, when a move is made, some moves are then barred for the next player. Overall, the rules are typically Carrollian-baroque.

Magician's Own Book. 1857.

The certain game, p. 243. As in Boy's Own Book.

The magical century, pp. 244-245. As in Boy's Own Book.

Book of 500 Puzzles. 1859.

The certain game, p. 57. As in Boy's Own Book.

The magical century, pp. 58-59. As in Boy's Own Book.

The Secret Out. 1859. Piquet on horseback, pp. 397-398 (UK: 130 131) -- additive (10, 100) unclearly explained.

Boy's Own Conjuring Book. 1860.

The certain game, pp. 213 214. As in Boy's Own Book.

Magical century, pp. 215. As in Boy's Own Book.

Vinot. 1860. Art. XI: Un cent de piquet sans cartes, pp. 19-20. (10. 100). Says the idea can be generalised, giving (7, 52) as an example.

Leske. Illustriertes Spielbuch für Mädchen. 1864? Prob. 563-III, pp. 247: Wer von 30 Rechenpfennigen den letzen wegnimmt, hat gewonnen. (6, 30).

F. J. P. Riecke. Mathematische Unterhaltungen. 3 vols., Karl Aue, Stuttgart, 1867, 1868 & 1873; reprint in one vol., Sändig, Wiesbaden, 1973. Vol. 3, art 22.2, p. 44. Additive form of (10, 100).

Mittenzwey. 1880. Probs. 286-287, pp. 52 & 101-102; 1895?: 315-317, pp. 56 & 103-104; 1917: 315-317, pp. 51 & 98.

(6, 30), last player wins.

(4, 15), last player loses, the solution discusses other cases: (7, 40), (7, 45) and indicates the general solution.

(added in 1895?) (3, 15), last player loses.

Hoffmann. 1893. Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300 301 = Hoffmann-Hordern, p. 197. (3, 15). c= Benson, 1904, The fifteen match puzzle, pp. 241 242.

Ball-FitzPatrick. 1st ed., 1898. Deuxième exemple, pp. 29-30. (6, 50).

E. Fourrey. Récréations Arithmétiques. (Nony, Paris, 1899; 2nd ed., 1901); 3rd ed., Vuibert & Nony, Paris, 1904; (4th ed., 1907); 8th ed., Librairie Vuibert, Paris, 1947. [The 3rd and 8th eds are identical except for the title page, so presumably are identical to the 1st ed.] Sections 65 66: Le jeu du piquet à cheval, pp. 48 49. Additive forms of (10, 100) and (7, 60). Then gives subtractive form for a pile of matches for (3, 17).

Étienne Ducret. Récréations Mathématiques. Garnier Frères, Paris, nd [not in BN, but a similar book, nouv. ed., is 1892]. Pp. 102 104: Le piquet à cheval. Additive version of (10, 100) with some explanation of the use of the term piquet. Discusses (6, 30).

Mr. X [possibly J. K. Benson -- see entry for Benson in Abbreviations]. His Pages. The Royal Magazine 9:3 (Jan 1903) 298-299. A good game for two. (3, 15) as a subtraction game.

David Devant. Tricks for Everyone. Clever Conjuring with Everyday Objects. C. Arthur Pearson, London, 1910. A counting race, pp. 52-53. (10, 100).

Dudeney. AM. 1917. Prob. 392: The pebble game, pp. 117 & 240. (3, 15) & (3, 13) with the object being to take an odd number in total. For 15, first player wins; for 13, second player wins. (Barnard (50 Telegraph ..., 1985) gives the case (3, 13).)

Blyth. Match-Stick Magic. 1921.

Fifteen matchstick game, pp. 87-88. (3, 15).

Majority matchstick game, p. 88. (3, 21).

Hummerston. Fun, Mirth & Mystery. 1924.

Two second-sight tricks (no. 2), p. 84. (6, 57), last player losing.

A match mystery, p. 99. (3, 21), last player losing.

H. D. Grossman & David Kramer. A new match-game. AMM 52 (1945) 441 443. Cites Dudeney and says Games Digest (April 1938) also gave a version, but without solution. Gives a general solution whether one wants to take an odd total or an even total.

C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. How to design a "Pircuit" or Puzzle circuit, pp. 388-394. On pp. 388-391, Harry Rudloe describes a relay circuit for playing the subtractive form of (3, 13), which he calls the "battle of numbers" game.

Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 24: "Ungerade" gewinnt, pp. 16 & 44 45. (= 'Odd' is the winner, pp. 18 & 53 55.) (7, 41) with the winner being the one who takes an odd number in total. Solves (7, b) and states the structure for (a, b).

I also have some other recent references to this problem. Lewis (1983) gives a general solution which seems to be wrong.